Sample Complexity of Solving Non-Cooperative Games

Abstract

This paper studies the complexity of solving two classes of non-cooperative games in a distributed manner, in which the players communicate with a set of system nodes over noisy communication channels. The complexity of solving each game class is defined as the minimum number of iterations required to find a Nash equilibrium (NE) of any game in that class with $\epsilon $ accuracy. First, we consider the class $\mathcal {G}$ of all $N$ -player non-cooperative games with a continuous action space that admit at least one NE. Using information-theoretic inequalities, a lower bound on the complexity of solving $\mathcal {G}$ is derived which depends on the Kolmogorov $2\epsilon $ -capacity of the constraint set and the total capacity of the communication channels. Our results indicate that the game class $\mathcal {G}$ can be solved at most exponentially fast. We next consider the class of all $N$ -player non-cooperative games with at least one NE such that the players’ utility functions satisfy a certain (differential) constraint. We derive lower bounds on the complexity of solving this game class under both Gaussian and non-Gaussian noise models. Finally, we derive upper and lower bounds on the sample complexity of a class of quadratic games. It is shown that the complexity of solving this game class scales according to $\Theta \left ({\frac {1}{\epsilon ^{2}}}\right)$ where $\epsilon $ is the accuracy parameter.